Hollow Module Research Articles

G-Small Intersection Graph of a Module

Let be a commutative ring with identity, and be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of are studied. Properties of such as connectedness, and completeness are considered. Besides, the girth and the diameter of are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph is complete if, is a generalized hollow module or is a direct sum of two simple modules, is proved.

L-Fuzzy Hollow Modules and L-Fuzzy Multiplication Modules

In this paper, we give some characterizations of L-fuzzy hollow modules and of L-fuzzy multiplication modules.

Purely Small Submodules and Purely Hollow Modules

The main goal of this paper is to give a new generalizations for two important classes in the category of modules, namely the class of small submodules and the class of hollow modules. They are purely small submodules and purely hollow modules respectively. Various properties of these classes of modules are investigated. The relationship between purely small submodules and P-small submodules which is introduced by Hadi and Ibrahim, is studied. Moreover, another characterization of purely hollow modules is considered.

Direct sum of various generalizations of h-lifting modules

In this work, [Formula: see text]-lifting modules, a special extension of [Formula: see text]-lifting which is a further generalization of lifting modules, are studied. Here, we observed that finite direct sum of [Formula: see text]-lifting modules may not be [Formula: see text]-lifting, so we provided various sufficient conditions for which [Formula: see text]-lifting modules are closed under direct sum. Moreover, we have introduced and studied the properties of a new version of [Formula: see text]-lifting module namely, [Formula: see text]-lifting, [Formula: see text]-lifting and completely [Formula: see text]-lifting modules and develop some more properties of the [Formula: see text]-lifting modules in terms of these modules. Further, we proved that if the direct sum of arbitrary family of hollow modules is [Formula: see text]-lifting, then the arbitrary direct sum of [Formula: see text]-lifting [Formula: see text]-module is again [Formula: see text]-lifting.

F-Hollow modules

In previous research, a concept was presented f –small submodules A submodule L of an R-module W is named f –small if L+X=W and W/X are small-singular, then X=W. In this work, we present a new concept, namely f –hollow modules, an R-module W is f-hollow if every submodule in W is an f-small submodule. This concept is an expansion of Hollow modules. Some characteristics are given in expansion to their relationship to a few of the concepts and the comparability between them is examined.

On Indeterminacy (Neutrosophic ) of Hollow Modules

In this paper, we formulate and study a new property, namely indeterminacy (neutrosophic) of the hollow module. We mean indeterminacy hollow module is neutrosophic hollow module B (shortly Ne(B)) such that it is not possible to specify the conditions for satisfying it. Some concepts have been studied and introduced, for instance, the indeterminacy local module, indeterminacy divisible module, indeterminacy indecomposable module and indeterminacy hollow-lifting module. Also, we investigate that if Ne(B) is an indeterminacy divisible module with no indeterminacy zero divisors, then any indeterminacy submodule Ne(K) of Ne(B) is an indeterminacy hollow module. Further, we study the relationship between the indeterminacy of hollow and indecomposable modules. Finally, conclusions and discussions for our obtained results are given.

Design and Characterization of a Hollow and Duplex-Chambered Propulsion Module for Narrow Pipe Inspection

Abstract Conventional inspection procedures using industrial endoscope cameras find it challenging to inspect narrow pipes because the endoscope cannot be inserted in the pipes due to the increasing friction and complex pipe structure. Active propulsion techniques can help improve inspection efficiency; however, the locomotion mechanism in such a narrow space is challenging because of the pipe’s geometrical configuration. In this study, a novel flexible propulsion module is proposed for narrow pipe inspection. The module is designed based on a newly proposed hollow-type duplex-chambered inchworm mechanism, which achieves smooth inchworm locomotion using only two air supply lines. With a simple attachable module design, the inspection operator can easily add active propulsion capabilities onto the regular industrial endoscope camera with a maximum diameter of 7 mm. The concept of the system and the structure of the mechanism are presented herein. Various experiments and simulations were performed to reveal the characteristics of the mechanism, and the results are discussed in terms of operating pressure, generated force, and locomotion speed, among other parameters. A prototype demonstrated smooth locomotion through 31-mm pipes at 26.6, 22.1, and 51.1 mm/s for horizontal locomotion, vertical climbing, and vertical descent, respectively, and performed well through a bending pipe.

E*-Essential Small Submodules and e*-Hollow Modules

The purpose of this paper is to introduce new concepts in a module  over a ring , the first one is called e*- small essential submodule, which is a generalization of the small submodule, the second concept is called e*-radical submodule which is a generalization of the radical submodule and the last concept is called e* - hollow module which is a generalization of the hollow module. We will prove some properties of all these concepts.

Loc-hollow Fuzzy Modules with Related Modules

The concept of a small f- subm was presented in a previous study. This work introduced a concept of a hollow f- module, where a module is said to be hollow fuzzy when every subm of it is a small f- subm. Some new types of hollow modules are provided namely, Loc- hollow f- modules as a strength of the hollow module, where every Loc- hollow f- module is a hollow module, but the converse is not true. Many properties and characterizations of these concepts are proved, also the relationship between all these types is researched. Many important results that explain this relationship are demonstrated also several characterizations and properties related to these concepts are given.

The Dual Notion of St-Polyform Modules

The concept of St-Polyform modules, was introduced and studied by Ahmed in [1], where a module M is called St-polyform, if for every submodule N of M and for any homomorphism 𝑓:N M; ker𝑓 is St-closed submodule in N. The novelty of this paper is to dualize this class of modules, the authors call it CSt-polyform modules, and according to this dualizations, some results which appeared in [1] are dualized for example we prove that in the class of hollow modules, every CSt-polyform module is coquasi-Dedekind. In addition, several important properties of CSt-polyform module are established, and other characterization of CSt-polyform is given. Moreover, many relationships of CSt-polyform modules with other related concepts are considered such as copolyform, epiform, CSt-semisimple, -nonsingular modules, and some others will be introduced such as non CSt-singular and G. coquasi-Dedekind modules.

Z-Small Submodules and Z-Hollow Modules

A submodule Ϝ of an R-module Ε is called small in Ε if whenever , for some submodule W of Ε , implies . In this paper , we introduce the notion of Ζ-small submodule , where a proper submodule Ϝ of an R-module Ε is said to be Ζ-small in Ε if , such that , then , where is the second singular submodule of Ε . We give some properties of Ζ-small submodules . Moreover , by using this concept , we generalize the notions of hollow modules , supplement submodules, and supplemented modules into Ζ-hollow modules, Ζ-supplement submodules, and Ζ-supplemented modules. We study these concepts and provide some of their relations .

Semi local hollow modules

To consider (R) is a commutative ring with unitary, (X) be a non--Zero unity left (R - module); (X) is known hollow module if each proper sub module of (X) is semi small local hollow module if (X) has a unique maximal submodule which contains each semi small submodule. The current study deals with this class of modules and give several fundamental properties related with this concept.

A Miniature Manipulator With Variable Stiffness Towards Minimally Invasive Transluminal Endoscopic Surgery

This letter presents a miniature manipulator with variable stiffness towards minimally invasive transluminal endoscopic surgery, such as the endoscopic submucosal dissection (ESD). The manipulator comprises hollow modules with holes in the sidewall, compact with a 4 mm diameter, and dexterous with six degrees of freedom (DOFs). Basic operations of grasping and stripping can be achieved via the terminal surgical tools. The flexible tubes and torque coils are adopted to avoid the motion coupling of the joints. The worm gears with an anti-backdrive property are used to transfer power from the motor to the reels. The variable stiffness is achieved by using the spring-sliding block-based driver system, delivering both high payload/accuracy and flexibility. The system is described in detail. The kinematics, workspace and variable stiffness are analyzed. Tests in terms of the grasping force and variable stiffness are conducted. Results show that the grasping force is larger than 3 N, and the manipulator's stiffness can be tuned with high rates of change. The manipulator's performances, such as the dexterity and the coordination of operation under the master-slave configuration, were preliminarily demonstrated through the basic mock-up operations.

On NM-Hollow Modules

This notion investigates the generalization of hollow module namely near maximal hollow module (NM-hollow module). Any module M is NM-hollow if for every H satisfy two properties namely nearly and maximal. Some new results have been studied and more relationships are presented. The main result in this section is; if M is a faithful and multiplication over the ring R, then M is NM-hollow module. Also, a direct-sum of NM-hollow module is also NM-hollow.

J-semi regular modules

Let R be a ring with identity and let M be a left R-module. M is called J-semiregular module if every cyclic submodule of M is J-lying over a projective summand of M, The aim of this paper is to introduce properties of J-semiregular module Especially, we give characterizations of J-semiregular module. On the other hand, the notion of J-semi hollow modules is studied as a generalization of semi hollow modules, finally F-J-semiregular modules is studied as a generalization of F-semiregular modules.

Lifting modules with finite internal exchange property and direct sums of hollow modules

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

Direct Sum and Projectivity of SemiHollow-Lifting Modules

Let ℝ be a ring with identity and let T be a unitary left Module over ℝ. In this paper, we give some cases when a direct sum of hollow Modules is semihollow-lifting, Also; we give a condition under which a direct sum of two Modules is semihollow-lifting,

On µ-Semiregular Module

Let R be an associative ring with identity and let M be right R-module M is called μ-semi hollow module if every finitely generated submodule of M is μ-small submodule of M The purpose of this paper is to give some properties of μ-semi hollow module. Also, we gives conditions under, which the direct sum of μ-semi hollow modules is μ-semi hollow. An R-module is said has a projective μ-cover if there exists an epimorphism f:P→M Where P is a projective R-module and ker (f)≪ P.And study some properties of Projective μ-cover of M. Were studied Moreover, An module M is μ-semiregular module if every cyclic submodule of M is μ-lying summand of M We add some results of μ-semiregular module.

L-Hollow modules

To consider R is a commutative ring with unity, be a nonzero unitary left R-module, is known hollow module if each proper submodule of is small. L-hollow module is a strong form of hollow module, where an R-module is known L-hollow module if has a unique maximal submodule which contains each small submodules. The current study deals with this class of modules and give several fundamental properties related with this concept.

On Jacobson – Small Submodules

Let R be an associative ring with identity and let M be a unitary left R–module. As a generalization of small submodule , we introduce Jacobson–small submodule (briefly J–small submodule ) . We state the main properties of J–small submodules and supplying examples and remarks for this concept . Several properties of these submodules are given . Also we introduce Jacobson–hollow modules ( briefly J–hollow ) . We give a characterization of J–hollow modules and gives conditions under which the direct sum of J–hollow modules is J–hollow . We define J–supplemented modules and some types of modules that are related to J–supplemented modules and introduce properties of this types of modules . Also we discuss the relation between them with examples and remarks are needed in our work.